Definition:Field (Abstract Algebra)
This page is about Field in the context of Abstract Algebra. For other uses, see Field.
Definition
A field is a non-trivial division ring whose ring product is commutative.
Thus, let $\struct {F, +, \times}$ be an algebraic structure.
Then $\struct {F, +, \times}$ is a field if and only if:
- $(1): \quad$ the algebraic structure $\struct {F, +}$ is an abelian group
- $(2): \quad$ the algebraic structure $\struct {F^*, \times}$ is an abelian group where $F^* = F \setminus \set 0$
- $(3): \quad$ the operation $\times$ distributes over $+$.
This definition gives rise to the field axioms, as follows:
Field Axioms
The properties of a field are as follows.
For a given field $\struct {F, +, \circ}$, these statements hold true:
\((\text A 0)\) | $:$ | Closure under addition | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x + y \in F \) | ||||
\((\text A 1)\) | $:$ | Associativity of addition | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle \paren {x + y} + z = x + \paren {y + z} \) | ||||
\((\text A 2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x + y = y + x \) | ||||
\((\text A 3)\) | $:$ | Identity element for addition | \(\displaystyle \exists 0_F \in F: \forall x \in F:\) | \(\displaystyle x + 0_F = x = 0_F + x \) | $0_F$ is called the zero | |||
\((\text A 4)\) | $:$ | Inverse elements for addition | \(\displaystyle \forall x: \exists x' \in F:\) | \(\displaystyle x + x' = 0_F = x' + x \) | $x'$ is called a negative element | |||
\((\text M 0)\) | $:$ | Closure under product | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x \circ y \in F \) | ||||
\((\text M 1)\) | $:$ | Associativity of product | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle \paren {x \circ y} \circ z = x \circ \paren {y \circ z} \) | ||||
\((\text M 2)\) | $:$ | Commutativity of product | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x \circ y = y \circ x \) | ||||
\((\text M 3)\) | $:$ | Identity element for product | \(\displaystyle \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:\) | \(\displaystyle x \circ 1_F = x = 1_F \circ x \) | $1_F$ is called the unity | |||
\((\text M 4)\) | $:$ | Inverse elements for product | \(\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:\) | \(\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x \) | ||||
\((\text D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z} \) |
These are called the field axioms.
Addition
The distributand $+$ of a field $\struct {F, +, \times}$ is referred to as field addition, or just addition.
Product
The distributive operation $\times$ in $\struct {F, +, \times}$ is known as the (field) product.
Also defined as
Some sources do not insist that the field product of a field is commutative.
That is, what they define as a field, $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as a division ring.
When they wish to refer to a field in which the field product is commutative, the term commutative field is used.
Also see
- Results about fields can be found here.
Linguistic Note
Note that while in English the word for this entity is field, its name in other European languages translates as body.
The original work was done by Richard Dedekind, who used the word Körper. When translating his work into English, Eliakim Moore mistakenly used the word field.
The translator into French did not make the same mistake.
Internationalization
Field is translated:
In Dutch: | lichaam | (literally: body) | ||
In Dutch: | veld | (literally: field) | ||
In French: | corps | (literally: body) | ||
In German: | Körper | (literally: body) | ||
In Russian: | Поле | (literally: field) | Pronounced: póh-lye | |
In Spanish: | campo | (literally: field) | ||
In Spanish: | cuerpo | (literally: body) |
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 23$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 14$. Definition of a Field
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields
- 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19$: Properties of $\Z_m$ as an algebraic system
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(4)$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0$ Zero
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0$ Zero
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: field